Not sure where you got those 1000 years from because they are not entirely correct.
Oh, I see that you have corrected it to 300 years. Fair enough.
Take a look at the link here. It's about the so-called Greater Iran. Made by Iranian contributors I assume, mainly. Then look at the maps of the various empires on the right. The light green are vassal states.
https://en.wikipedia.org/wiki/Greater_Iran
It was only the Achaemenid Empire, as I told, that had any real control/influence of Levant and that was not for more than 250 years. But Iranian rulers left the locals alone and used vassals most of the time.
The only Arab country of today that you had a very long presence in is Iraq. Other than that there is not really any other Arab country.
Syria, Lebanon etc. are all fairly new political alliances.
Being reckoned with I agree. After all Iran is not a small country. But currently there are many social and political conflicts. For each year I think that the differences in views inside Iran are only becoming more bigger and extreme. If the Iranian people had a tradition of rebelling against their rulers and been lightly armed then I think we would have seen similar "Arab Springs" in Iran or at least some provinces.
I don't think that the Mullah's will last much longer. An foreign attack would remove them overnight.
At least other regimes in the region are not as reactionary in terms of foreign politics. Every sane player on the international scene would have conducted themselves much differently than Iran has since 1979.
If you had normalized relations you might have reached the levels of GCC in terms of standard of livings and economy although Iran has a population of 75 million and they will never be as rich nor their citizens as some of the smaller Gulf States just due to the demographics.
Iranian influence is getting smaller and smaller. Just 7 years ago HizbAlShaitan had a tremendous support among all Arabs regardless of country, sect etc. Even huge trucks of humanitarian aid reached HizbAlShaitan and Lebanon from the GCC. Same with Palestine were Hamas turned their backs towards you after the HizbAlShaitan interference in Syria against their fellow Sunni Arab brethren.
Then look at the situation today. Houthi's irrelevant in Yemen again (Iranian support there was highly limited but let us just include it for the sake of it), Iraq is getting more and more independent and stronger for each year. The Bahraini revolution was a fiasco and now Bahrain is more or less a province of KSA de facto. Syria is falling apart. From a stable ally to a war-ridden country that is uncontrollable in large parts. HizbAlShaitan is losing more and more support in Lebanon while we speak.
I don't really see any progress, in fact I see decrease. The Iranian Mullah's also failed to take advantage of the protests in the Eastern Province.